In our paper, we consider the \(q\) -deformed totally asymmetric zero range process (\(q\) -TAZRP), of which a definition is given below. This model was considered by Borodin and Corwin as a specialization of Macdonald processes, and then by Borodin, Corwin and Sasamoto in [1]} in the light of its stochastic duality (in the sense of [2]}) to the \(q\) -TASEP model. Our main inspiration is the paper [3]} by Korhonen and Lee, where the transition probability for the \(q\) -TAZRP with homogeneous jump rates is derived. In [3]} the authors used Tracy and Widom's method rather than results from Macdonald processes. Our paper is a generalization to [3]}, and we extend their method to inhomogeneous jump rates. Let us note that, this integrable ZRP was first introduced by Sasamoto and Wadati [6]} under the name of \(q\) -Boson totally asymmetric diffusion model, and this name is used in later publications like [7]}, so we refer to this name in the title. But later in this paper we follow the nomenclature in [3]} and call the model \(q\) -TAZRP.
